book excerptise:   a book unexamined is wasting trees

Mathematical Recreations and Essays

Harold Scott Macdonald Coxeter and Walter William Rouse Ball

Coxeter, Harold Scott Macdonald; Walter William Rouse Ball;

Mathematical Recreations and Essays

Courier Dover Publications, 1987, 448 pages

ISBN 0486253570, 9780486253572

topics: |  math | puzzle | history | geometry |



Fifth Method for guessing a number (Modular Arithmetic)

	 			 p.6 [source: Bachet, problem VI, p. 84]

   Ask anyone to select a number n less than 60. Request him to perform the
   following operations.

	  (i) Divide by 3 and mention the remainder; say a.
	 (ii) Divide by 4, remainder b;
	(iii) divide by 5, remainder c.

    Then the number selected is the remainder obtained by dividing 40a + 45b
    + 36c by 60.

[Why this works: 

 Let:  A = 40 = multiple of 4,5, that is 1 more than some multiple of 3;
	 B = 45 = multiple of 3,5, that is 1 more than some multiple of 4;
	 C = 36 = multiple of 3,4, +1 of some multiple of 5;
		 [can have more than three multiples; their product p must be
			 > chosen number n]
	 p = 60 = product of 3,4,5...
	 N = Aa + Bb + Cc + ...

now, 40%3 =1, so 40a%3 = a.  
     45b, 36c etc are divisible by 3.  
     Hence N = Aa + Bb + Cc divided by p has remainder = a
     Since n % 3 is also = a, 
     hence, N-n is divisible by 3,4,5... hence by p.  
     since p>n, N % p = n


[summary of the general argument]

Let a', b', c', ... be a series of numbers prime to one another, such that
their product p is > the chosen number n.

Let a, b, c, ... be the remainders when n is divided by a', b', c',
Find a number A which is a multiple of the product b'c'd'...
and which is +1 of a multiple of a'. Find a number B which is a
multiple of a'c'd' ... and which exceeds by +1 a multiple of b', and find
analogous numbers C, D, ...

modulo a', b', c'-

in general, if the numbers a', b', c' are small, the corresponding
numbers A, B, C can be found by inspection.

I proceed to show that n is equal to the remainder when Aa + Bb + Cc + ... is
divided by p.

Let N = Aa + Bb + Cc + ...

Now A % a' = 1, therefore Aa % a' = a.
The other terms, Bb, Cc etc. are divisible by a'.

Hence, if N is divided by a', the remainder is a. Also if n is divided by a',
the remainder is a.

Therefore N - n is divisible by a', b', ... - hence by p.  Hence N = kp + n.
Since n is less than p, hence if N is divided by p, the remainder 18 n.

In the above example from Bachet, a',b', c' = 3,4,5.  So p = 60,
By inspection, A = 40 [multiple of 4*5, s.t. A-1 is divisible by 3]
B = 45, C = 36.


EXAMPLE 1:
3,4,7 (<84) : N = (28a + 21b + 36c) % p

guess 33: 0,1,5 : N = 0+ 21 + 180 = 201 % 84 = 33

EXAMPLE 2 (for all 2-digit nums):  3,5,7 (p=105)-> 70.a + 21.b + 15.c
guess 33: 0,3,5 -> N=0 + 63 + 75 = 138 % 105 = 33
guess 47 : 2,2,5 -> N= 140 + 42 + 75 %105 = 35 + 12 = 47



13. Calculating Prodigies

Zerah Colburn (1804-1840)


son of a small farmer in Cabut, Vermont, USA.

While still less than six years old he showed extraordinary powers of mental
calculation, which were displayed in a tour in America.  Two years later he
was brought to England, where he was repeatedly examined by competent
observers. He could instantly give the product of two numbers each of four
digits, but hesitated if both numbers exceeded 10,000. Among questions asked
him at this time were to raise 8 to the 16th power; in a few seconds he gave
the answer 281,474,976,710,656, which is correct. He was next asked to raise
the numbers 2, 3, ... 9 to the lOth power: and he gave the answers so rapidly
that the gentleman who was taking them down was obliged to ask him to repeat
them more slowly; but he worked less quickly when asked to raise numbers of
two digits like 37 or 59 to high powers. He gave instantaneously the square
roots and cube roots (when they were integers) of high numbers, e.g. the
square root of 106,929 and the cube root of 268,336,125; such integral roots
can, however, be obtained easily by various methods.

More remarkable are his answers to questions on the factors of numbers. Asked
for the factor of 247.4SS. he replied 941 and 263; asked for the factors of
171,395, he gave 5, 7, 59, and 83; asked for the factors of 36,083, he said
there were none. He, however, found it difficult to answer questions about
the factors of numbers higher than 1,000,000. His power of factorizing high
numbers was exceptional, and depended largely on the method of
two-digit terminals...

His English and American friends, raised money for his education, and he
was sent in succession to the Lycee Napoleon in Paris and Westminster
School in London. With education his calculating powers fell off, and he
lost the frankness which when a boy had charmed observers.


George Parker Bidder, 1806-1878

 							p.357

Bidder was born in 1806 at Moreton Hampstead, Devonshire, where his father
was a stone-mason. At the age of six he was taught to count up to 100, but
though sent to the village school, learnt little there, and at the beginning
of his career was ignorant of the meaning of arithmetical terms and of
numerical symbols. Equipped solely with this knowledge of counting, he taught
himself the results of addition, subtraction, and multiplication of numbers
(less than 100) by arranging and rearranging marbles, buttons, and shot in
patterns. In after-life he attached great importance to such concrete
representations, and believed that his arithmetical powers were strengthened
by the fact that at that time he knew nothing about the symbols for
numbers.  p.358

Seven years old: he heard a dispute between two of his neighbours about the
	price of something which was being sold by the pound, and to their
	astonishment remarked that they were both wrong, mentioning the
	correct price. After this exhibition the villagers delighted in
	trying to pose him with arithmetical problems.

Nine years old: His father  started to take him around to fairs, exhibiting
	his powers.  Some questions :

	What is the interest on £11,111 for 11,111 days at 5 per cent. a
	year 1 answer, £16,911 11s., given in one minute.  How many hogsheads
	of cider can be made from a million of apples, if 30 apples make one
	quart 1 answer, 132 hogsheads 11 gallons] quart and 10 apples over,
	given in 35 seconds...

  	At age 11, some Cambridge mathematicians raised a fund and started
	his education.  But a few months later, his father took him out
	again as a showman.

Age 14 (1819): In addition to answering questions on products of numbers
	and the number of specified units in given quantities, he was, after
	1819, ready in finding square roots, cube roots, etc., of high
	numbers, it being assumed that the root is an integer.  [many others
	also, e.g.  Colburn]

	Qs: Find two numbers whose difference is 12 and whose product,
	multiplied by their sum, is equal to 14,560?  answer, 14 and 26.
	Find a number whose cube less 19 multiplied by its cube shall be
	equal to the cube of 6: answer, 3, given instantly.

Eventually at Edinburgh university, some members persuaded his father to
let him study, and he graduated as a Civil Engineer.  He rose to high
distinction in the profession. 358

In 1818, in the course of a tour, young Bidder was pitted against Colburn
(ages 13 and 14), and on the whole proved the abler calculator.   358

In factorizing numbers he was less successful than Colburn, and was generally
unable to deal at sight with numbers higher than 10,000. 359

Career

Bidder retained his power of rapid mental calculation to the end of his life,
and as a constant parliamentary witness in matters connected with engineering
it proved a valuable accomplishment. Just before his death an illustration of
his powers was given to a friend who, talking of then recent discoveries,
remarked that if 36,918 waves of red light which only occupy one inch are
required to give the impression of red, and if light travels at 190,000 miles
a second, how immense must be the number of waves which must strike the eye
in one second to give the impression of red. "You need not work it out," said
Bidder; "the number will be 444,433,651,200,000." 362

members of the Bidder family have also shown exceptional powers of a similar
kind as well as extraordinary memories.  Of Bidder's elder brothers, one
became an actuary, and on his books being burnt in a fire, he rewrote them in
six months from memory, but, it is said, died of consequent brain fever

Bidder's eldest son, a lawyer of eminence, was able to multiply together two
numbers each of fifteen digits.  Neither in accuracy nor rapidity was he
equal to his father, but, then, he never steadily and continuously devoted
himself to developing his abilities in this direction. He remarked that in
his mental arithmetic, he worked with pictures of the figures, and said, "If
I perform a sum mentally it always proceeds in a visible form in my mind;
indeed I can conceive no other way possible of doing mental arithmetic ":
this, it will be noticed, is opposed to his father's method. Two of his
children, one son and one daughter, representing a third generation,
inherited analogous powers. p.362

Jacques Inaudi


Jacques Inaudi was born in 1867 at Onorato in Italy. Be was employed in early
years as a shepherd, and spent the long idle hours in which he had no active
duties in pondering on numbers, but used for them no concrete representations
such as pebbles.  366

His calculating powers first attracted notice about 1873. Shortly afterwards
his elder brother sought his fortune as an organ-grinder in Provence, and
young Inaudi, accompanying him, came into a wider world, and earned a few
coppers for himself by street exhibitions of his powers.  In 1880 he visited
Paris, where he gave exhibitions... He was then ignorant of reading and
writing: these arts he subsequently acquired.

A performance before the general public rarely lasted more than 12
minutes... A normal programme included the subtraction of one number of
twenty-one digits from another number of twenty-one digits: the addition of
five numbers each of six digits: the multiplying of a number of four digits
by another number of four digits : the extraction of the cube root of a
number of nine digits, and of the fifth root of a number of twelv..e digits:
the determination of the number of seconds in a period of time, and the day
of the week on which a given date falls.

Of course, the questions were put by members of the audience.  To a
professional calculator these problems are not particularly difficult.  As
each number was announced, Inaudi repeated it slowly to his assistant, who
wrote it on a blackboard, and then slowly read it aloud to make sure that it
was right. Inaudi then repeated the number once more. By this time he had
generally solved the problem, but if he wanted longer time he made a few
remarks of a general character, which he was able to do without interfering
with his mental calculations. Throughout the exhibition he faced the
audience: the fact that he never even glanced at the blackboard added to the
effect.


Synesthesia

Bidder did not visualize a number like 984 in symbols, but thought of it in
a concrete way as so many units which could be arranged in 24 groups of 41
each.

It should also be observed that he, like Inaudi, relied largely on the
auditory sense to enable him to recollect numbers.  "For my own part," he
wrote, in later life, "though much accustomed to see sums and quantities
expressed by the usual symbols, yet if I endeavour to get any number of
figures that are represented on paper fixed in my memory, it takes me a
much longer time and a very great deal more exertion than when they are
expressed or enumerated verbally." For instance, suppose a question put to
find the product of two numbers each of nine digits, if they were "read to
me, I should not require this to be done more than once; but if they were
represented in the usual way, and put into my hands, it would probably take
me four times to peruse them before it would be in my power to repeat them,
and after all they would not be impressed so vividly on my imagination. "
... 361-2

It is probable that the majority of oalculating prodigies rely on the
speech-muscles as well as on the eye and the ear to help them to recollect
the figures with which they are dealing. It was formerly believed that they
all visualized the numbers proposed to them, and certainly some have done
so. Inaudi, however, trusted mainly to the ear and to articulation. Bidder
also relied partly on the ear, and when he visualized a number, it was not as
a coIlection of digits, but as a concrete coIlection of units divisible, if
the number was composite, into definite groups. Ruckle relied mainly on
visualizing the numbers. 367-8

So it would seem that there are different types of the memories of
calculators. Inaudi could reproduce mentaIly the sound of the repetition of
the digits of the number in his own voice, and was confused, rather than
helped, if the numbers were shown to him in writing.

The articulation of the digits of the number also seemed necessary to enable
him fully to exhibit his powers, and he was accustomed to repeat the numbers
aloud before beginning to work on them - the sequence of sounds being
important.

Explanation of their powers

Numbers were ever before Inaudi : he thought of little else, he dreamt of
them, and sometimes even solved problems in his sleep. His memory was
exceIlent for numbers, but normal or subnormal for other things.  At the end
of a seance he could repeat the questions which had been put to him and his
answers, involving hundreds of digits in all. Nor was his memory in such
matters limited to a few hours. Once, eight days after he had been given a
question on a number of twenty-two digits, he was unexpectedly asked about
it, and at once repeated the number. He was repeatedly examined, and we know
more of his work than of any of his predecessors, with the possible exception
of Bidder. 368

Bidder gave an account of the processes he had discovered and used in a
lecture given by him in 1856 to the Institution of Civil Engineers.  [Ball
remarks that these may be his perfected analysis rather than may not be how
he actually did them, esp as a child; modern cognitive scientists like
Ericsson would completely agree.] 369


Modern Prodigies: Shakuntala Devi


[see Barlow, F. Mental prodigies. (1952). , and
Smith, S.B. The great mental calculators (1983)]

	[by Arthur Jensen, School of Education, UC Berkeley: "Speed of
	Information Processing in a Calculating Prodigy"; INTELLIGENCE,
	v.14, 259-274 (1990) Material added for relevance]

Shakuntala Devi was born in Bangalore, India, in 1940, to a 15-year-old
mother and 61-year-old father, who was a circus acrobat and magician. Devi
travelled with him since she was 3, performing card tricks, from which she
cultivated her facility with numbers.  Her talent in this sphere was
manifested early; at age 5 she could already extract cube roots quickly in
her head, and she soon began supporting herself and the rest of her family as
a stage performer, travelling throughout India billed as a calculating
prodigy. Even before she was in her teens, she began travelling around the
world, performing numerical feats, usually before audiences in colleges and
universities.

In addition to mathematics texts such as Figuring: The joy of numbers
(1978),  Shakuntala Devi has also written a book on
The world of homosexuals (1977).

Mathematical demonstrations


It seems hard to believe, but the following is reported in the Guinness Book of
Records (1982), which has a reputation for the authenticity of its claims:

	Mrs.  Shakuntala Devi of India demonstrated the multiplication of two
	13-digit numbers of 7,686,369,774,870 x 2,465,099,745,779 picked at
	random by the Computer Department of Imperial College, London on 18
	June 1980, in 28 s. Her correct answer was
	18,947,668,177,995,426,462,773,730.

An article in the New York Times (November 10, 1976, cited in Smith, 1983,
p. 306) reported that Shakuntala Devi added the following four numbers and
multiplied the result by 9,878 to get the (correct) answer 5,559,369,456,432:

		25,842,278
		111,201,721
		370,247,830
		55,511,315

She was reported to have done this calculation in "20 seconds or less."  At
Southern Methodist University, in 1977, Devi extracted the 23rd root of a
201-digit number in 50 s. Her answer--546,372,891--was confirmed by
calculations done at the U.S. Bureau of Standards by the Univac 1101
computer, for which a special program had to be written to perform such a
large calculation (Smith, 1983).

Stanford demonstration 1988


in 1988, Devi visited the San Francisco Bay Area, when I had the opportunity
to observe a demonstration she gave at Stanford University before an audience
filled with mathematicians, engineers, and computer experts, who had come
with their electronic calculators or printouts of large problems that had
been submitted to the University's main-frame computer.

I was curious, fast of all, to see if Devi had the kind of autistic
personality so commonly associated with such unusual mental feats. Also, I
wanted to measure her performance times myself, to see if they substantiated
the astounding claims I had read of her calculating prowess. But mainly, if
the claims proved authentic, I hoped I could persuade her to come to Berkeley
to be tested in my chronometric laboratory, so we could measure her basic
speed of information processing on a battery of elementary cognitive tasks
(ECTs) for which the results could be compared with the reaction time (RT)
data we had obtained on the same ECTs in large samples of students and older
adults. Indeed, Devi kindly consented to come to my laboratory and spent
about 3 h taking various tests. In addition, she spent some 2 h with me,
discussing her life and work.

At her Stanford appearance, Shakuntala Devi, in a colorful silk sari, sat at
a table in front of the blackboard in a lecture hall. The demonstration
lasted almost 90 rain. (Engaging in such intense mental activity beyond that
length of time, Devi said, she begins to feel tired.) Problems involving
large numbers were written on the blackboard by volunteers from the audience,
many of whom knew of Devi's reputation and had brought along computer
printouts with the problems and answers. Devi would turn around to look at a
problem on the blackboard, and always in less than 1 rain (but usually in
just a few seconds) she would state the answer, or in the case of solutions
involving quite large numbers she would write the answer on the blackboard.

[AJ and his wife are on row 1, measuring her response time with a stopwatch.
Answers are always correct, and it almost never takes more than a minute.  ]

When I handed Devi two problems, each on a separate card, thinking she would
solve first one, then the other, my wife was taken by surprise, as there was
hardly time to start the stopwatch, so quick was Devi's response. Holding the
two cards side-by-side, Devi looked at them briefly and said, "The answer to
the first is 395 and to the second is 15. Right?" Right, of course! (Her
answers were never wrong.) 

Handing the cards back to me, she requested that I read the problems aloud to
the audience. They were: (a) the cube root of 61,629,875 (= 395), and (b) the
7th root of 170,859,375 (= 15). I was rather disappointed that these problems
seemed obviously too easy for Devi, as I had hoped they would elicit some
sign of mental strain on her part. After all, it had taken me much longer to
work them with a calculator.

	cuberoot of a 10-digit number (13343) : 10s
	8th root of 14-digit number (468) : 10s
	7th root of 33-digit number (462957) : 40s

Devi also does noninteger exact fractional roots almost as fast as integer
roots-- averaging about 3 to 4 s longer... [not irrationals].

Apparently she does not apply a standard algorithm uniformly to every problem
of a certain type, such as square roots, or cube roots, or powers. Each
number uniquely dictates its own solution, so to speak.

The presence of commas only interferes with the "natural" (and virtually
automatic) dissolution of the number in Devi's mind. I have since learned
from an Indian professor that commas are not used in India's number system,
and it seems likely that their interfering effect for Devi could stem in part
from her intensive childhood experience in working with large numbers lacking
commas or any other form of triplet grouping.

Calendar calculations: To determine if anything besides sheer calculation
enters Devi's thought process while she is doing calendrical calculations, I
called out "January 30, 1948," to which she instantly answered, "That was a
Friday-- and the day that our great leader Mahatma Gandhi was assassinated."

Obviously her calendrical calculating does not entirely usurp her other
memory or thought processes.

Personal Characteristics. The first thing most observers would notice about
Devi is that her general appearance and demeanor are quite the opposite of
the typical image of the withdrawn, obsessive, autistic savant, so well
portrayed by Dustin Hoffman in the recent motion picture, Rain Man. Devi
comes across as alert, extraverted, affable, and articulate. Her English is
excellent, and she also speaks several other languages. She has the stage
presence of a seasoned performer, and maintains close rapport with her
audience. At an informal reception after her Stanford performance, I noticed
that among strangers she was entirely at ease, outgoing, socially adept,
self-assured, and an engaging conversationalist.  To all appearances, the
prodigious numerical talent resides in a perfectly normal and charming
lady. She is divorced and has a daughter attending college in England, who,
Devi remarks with mock dismay, uses a computer in her science and math
courses. In fact, Devi claims none of her relatives has ever shown any
mathematical talent.



Factorization based on Two-digit terminals


method used by Bidder (and Colburn): when it was exactly divisible.

for example, in dividing (say) 25,696 by 176, he first argued that the answer
must be a number of three digits, and obviously the left-hand digit must be
1.  Next he had noticed that there are only 4 numbers of two digits (namely,
21, 46, 71, 96) which when multiplied by 76 give a number which ends in
96. Hence the answer must be 121, or 146, or 171, or 196; and experience
enabled him to say without calculation that 121 was too small and 171 too
large.  Hence the answer must be 146.  If he felt any hesitation, he mentally
multiplied 146 by 176 (which he said he could do "instantaneously"), and
thus checked the result.

It is noticeable that when Bidder, Colburn, and some other calculating
prodigies knew the last two digits of a product of two numbers, they also
knew, perhaps subconsciously, that the last two digits of the separate
numbers were necessarily of certain forms.


Johann Martin Zacharias Dase (b.1824 Hamburg-1861)


Multiply 79,532,853 by 93,758,479: asked by Schumacher, answered in 54
seconds. In answer to a similar request to find the product of two numbers
each of twenty digits, he took 6 minutes; to find the product of two numbers
each of forty digits, he took 364 MATHEMATICAL RECREATIONS AND ESSAYS [cn.
40 minutes; to find the product of two numbers each of a hundred digits, he
took 8 hours 45 minutes. Gauss thought that perhaps on paper the last of
these problems could be solved in half this time by a skilled computator.

Like all these calculating prodigies, he had a wonderful memory, and an hour
or two after a performance could repeat all the numbers mentioned in it. Be
had also the peculiar gift of being able after a single glance to state the
number (up to about 30) of sheep in a Hock, of books in a case, and so on;
and of yisualizing and recolle£ting a large number of objects.  For instance,
after a second's look at some dominoes he gave the sum (1 17) of their
points; asked how many letters were in a certain line of print chosen at
random in a quarto page, he instantly gave the correct number (63); shown
twelve digits, he had in half a second memorized them and their positions so
as to be able to name instantly the particular digit occupying any assigned
place.

Dase - cooperated with mathematicians - compiled table of natural logarithms
- computed the first 1,005,000 numbers to 7 places of decimals in his
spare time from 1844 to 1847, when occupied by the Prussian survey.


Contents

Review of chapters from:
J. S. Frame, Bull. Amer. Math. Soc. Volume 46, Number 3 (1940), 211-213.

 1. ARITHMETICAL RECREATIONS
    includes arithmetical recreations whose interest is mainly
    historical rather than arithmetical. Some of these are of the "think of a
    number" type, others involve digit notations, and still others are tricks
    with cards or games with counters.
    TOC:
    	  Bachet, Ozanam, Montucla |
        To find a number selected by someone | 
        Prediction of the result of certain operations | 
        Problems involving two numbers | Digit questions |
	  Problems depending on the scale of notation | 
	  Other problems with numbers in the denary scale | 
	  Four fours problem | Problems with a series of numbered things | 
	  Arithmetical restorations | Calendar problems |
        Medieval problems in arithmetic | The Josephus problem. Decimation |
        Nim and similar games | Moore's game | Kayles | Wythoff's game |
        Addendum on solutions

 2. ARITHMETICAL RECREATIONS (continued)
    problems of probability derangements and arrangements, decimal
    expansions, rational triangles, finite arithmetics, D. H. Lehmer's number
    sieve for prime factors, and concludes with a discussion of p'erfect
    numbers, Mersenne's numbers, and Fermais theorem.
        Arithmetical fallacies | Paradoxical problems | 
	  Probability problems | Permutation problems | 
	  Bachet's weights problem | The decimal expression for 1/n |
        Decimals and continued fractions | Rational right-angled triangles |
	  Triangular and pyramidal numbers | Divisibility | 
        The prime number theorem | Mersenne numbers | Perfect numbers |
        Fermat numbers | Fermat's Last Theorem | Galois fields

 3. GEOMETRICAL RECREATIONS
    mainly geometrical fallacies and paradoxes, problems in dissection,
    cyclotomy and area-covering. The deltoid solution to Kakeya's minimal
    problem, erroneously attributed to Kakeya (p. 100) was really suggested
    by Professors Osgood and Kabota according to Question 39, American
    Mathematical Monthly (1921), p. 125.
        Geometrical fallacies | Geometrical paradoxes | 
        Continued fractions and lattice points | Geometrical dissections |
        Cyclotomy | Compass problems | The five-disc problem | 
        Lebesgue's minimal problem | Kakeya's minimal problem 99 | 
	  Addendum on a solution

 4. GEOMETRICAL RECREATIONS (continued)
    statical and dynamical games of position.  Among topics discussed are
    some extensions of the game of three in a row, tessellations of the
    plane, problems with moving counters, and the effect of cutting a Möbius
    strip in various ways.
        Statical games of position | Three-in-a-row. Extension to p-in-a-row
        | Tessellation | Anallagmatic pavements | Polyominoes | Colour-cube
        problem | Squaring the square | Dynamical games of position |
        Shunting problems | Ferry-boat problems | Geodesic problems |
        Problems with counters or pawns | Paradromic rings | Addendum on
        solutions

 5. POLYHEDRA
    comprehensive elementary discussion of the relations between the faces,
    edges, and vertices and the associated angles of the regular solids and
    the Archimedean solids, which is well illustrated by good
    figures. Stellated polyhedra, solid tessellations, and the kaleidoscope
    each receive some attention. The use of the term Platonic for the regular
    solids might be questioned since they were known before Plato.
        Symmetry and symmetries | The five Platonic solids | Kepler's
        mysticism | Pappus, on the distribution of vertices | Compounds | The
        Archimedean solids | Mrs. Stott's construction | Equilateral
        zonohedra | The Kepler-Poinsot polyhedra | The 59 icosahedra | Solid
        tessellations | Ball-piling or close-packing | The sand by the
        sea-shore | Regular sponges | Rotating rings of tetrahedra | The
        kaleidoscope

 6. CHESS-BOARD RECREATIONS
    recreations associated with the chessboard and with magic
    squares. Similar problems with dominoes and with magic cubes are also
    discussed.
        Relative value of pieces | The eight queens problem | Maximum pieces
        problem | Minimum pieces problem | Re-entrant paths on a chess-board
        | Knight's re-entrant path | King's re-entrant path | Rook's
        re-entrant path | Bishop's re-entrant path | Routes on a chess-board
        | Guarini's problem | Latin squares | Eulerian squares | Euler's
        officers problem | Eulerian cubes

 7. MAGIC SQUARES
        Magic squares of an odd order | Magic squares of a singly-even order
        | Magic squares of a doubly-even order | Bordered squares | Number of
        squares of a given order | Symmetrical and pandiagonal squares |
        Generalization of De la Loubere's rule | Arnoux's method |
        Margossian's method | Magic squares of non-consecutive numbers |
        Magic squares of primes | Doubly-magic and trebly-magic squares |
        Other magic problems | Magic domino squares | Cubic and octahedral
        dice | Interlocked hexagons | Magic cubes

 8. MAP-COLOURING PROBLEMS
    general theory of the four-colour problem more elaborately than the
    earlier editions of this book, mentions briefly such matters as
    orientable surfaces and dual maps, and more fully the seven-colour
    mapping problem on the torus, and finally considers various colouring
    problems on the regular polyhedra.
        The four-colour conjecture | The Petersen graph | Reduction to a
        standard map | Minimum number of districts for possible failure |
        Equivalent problem in the theory of numbers | Unbounded surfaces |
        Dual maps | Maps on various surfaces | Pits, peaks, and passes |
        Colouring the icosahedron

 9. UNICURSAL PROBLEMS
    mazes and other similar problems whose solutions depend on the unicursal
    tracing of a route through prescribed points (nodes) over various given
    paths.
        Euler's problem | Number of ways of describing a unicursal figure |
        Mazes | Trees | The Hamiltonian game | Dragon designs

10. COMBINATORIAL DESIGNS
    certain combinatorial problems known under the title of Kirkman's
    school-girl problems, and ends with a similar problem about arranging
    members of a bridge club at tables so that different members shall play
    together in successive rubbers.
        A projective plane | Incidence matrices | An Hadamard matrix | An
        error - correcting code | A block design | Steiner triple systems |
        Finite geometries | Kirkman's school-girl problem | Latin squares |
        The cube and the simplex | Hadamard matrices | Picture transmission |
        Equiangular lines in 3-space | Lines in higher-dimensional space |
        C-matrices | Projective planes

11. MISCELLANEOUS PROBLEMS
    the Fifteen Puzzle, the Tower of Hanoï, Chinese Rings, and various
    mathematical card
	The fifteen puzzle | The Tower of Hanoi | Chinese rings | Problems
        connected with a pack of cards | Shuffling a pack | Arrangements by
        rows and columns | Bachet's problem with pairs of cards | Gergonne's
        pile problem | The window reader | The mouse trap. Treize

12. THREE CLASSICAL GEOMETRICAL PROBLEMS
    famous classical problems concerning the duplication of the cube,
    trisection of an angle, and quadrature of the circle.
        The duplication of the cube [ Solutions by Hippocrates, Archytas,
              Plato, Menaechmus, Apollonius, and Diocles; Solutions by Vieta,
              Descartes, Gregory of St. Vincent, and Newton ]
        | The trisection of an angle
              [ Solutions by Pappus, Descartes, Newton, Clairaut, and Chasles ]
        | The quadrature of the circle | Origin of symbol it | Geometrical
        methods of approximation to the numerical value of Pi | Results of
        Egyptians, Babylonians, Jews | Results of Archimedes and other Greek
        writers | Results of European writers, 1200-1630 | Theorems of Wallis
        and Brouncker | Results of European writers, 1699-1873 |
        Approximations by the theory of probability

13. CALCULATING PRODIGIES
    calculating prodigies which introduces over a dozen famous mental
    calculators beginning with Jedediah Buxton and Thomas Fuller in the
    eighteenth century, and including two American calculators Zerah Colburn
    and Trueman Henry Safford, and gives something of their histories and the
    type of problems they could solve.
              John Wallis, 1616-1703 ; Buxton, circ. 1707-1772 ; Fuller,
              1710-1790; Amp,6re ; Gauss, Whately ; Colburn,1804-1840 ;
              Bidder, 1806-1878 ; Mondeux, Mangiamele ; Dase, 1824-1861 ;
              Safford, 1836-1901 ; Zamebone, Diamandi, Ruckle ; Inaudi, 1867- ]
        | Types of memory of numbers
        | Bidder's analysis of methods used
              [ Multiplication ; Digital method for division and factors ;
              Square roots. Higher roots ; Compound interest ; Logarithms ]
        | Alexander Craig Aitken

14. CRYPTOGRAPHY AND CRYPTANALYSIS
    cryptography and cryptanalysis written by Dr.  Abraham Sinkov. It
    presents in easily understandable form the chief elements in a
    cryptographic system, and gives various possible ways for attempting to
    solve such a cipher.
        Cryptographic systems | Transposition systems | Columnar
        transposition | Digraphs and trigraphs | Comparison of several
        messages | The grille | Substitution systems | Tables of frequency |
        Polyalphabetic systems | The Vigenere square | The Playfair cipher |
        Code | Determination of cryptographic system | A few final remarks
 

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This review by Amit Mukerjee was last updated on : 2015 Oct 10